MATHEMATICAL SYMBOLS

=

Equal sign: equality

≠

Not equal sign: inequality

≈

Approximately equal: approximation

>

Strict inequality: greater than

<

Strict inequality: less than

≥

Inequality: greater than or equal to

≤

Inequality: less than or equal to

<<

Inequality: much less than

>>

Inequality: much greater than

( )

Parentheses: calculate expression inside first

[ ]

Brackets: calculate expression inside first

+

Plus sign: addition

−

Minus sign: subtraction

±

Plus / minus: both plus and minus operations

*

Asterisk: multiplication

×

Times sign: multiplication

⋅

Multiplication dot: multiplication

÷

Division sign / obelus: division

/

Division slash: division

—

Horizontal line: division / fraction

mod

Modulo: remainder calculation

.

Period: decimal point / decimal separator

a^b

Power: exponent

a^b

Caret: exponent

√a

Square root: √a ⋅ √a = a

³√a

Cube root: ³√a ⋅ ³√a ⋅ ³√a = a

⁴√a

Fourth root: ⁴√a ⋅ ⁴√a ⋅ ⁴√a ⋅ ⁴√a = a

ⁿ√a

N-th root (radical)

%

Percent: 1% = 1/100

‰

Per-mille: 1‰ = 1/1000 = 0.1%

ppm

Per-million: 1ppm = 1/1000000

ppb

Per-billion: 1ppb = 1/1000000000

ppt

Per-trillion: 1ppt = 10^-12

GEOMETRY SYMBOLS

∠

Angle: formed by two rays

∟

Right angle: 90°

°

Degree: 1 turn = 360°

deg

Degree: 1 turn = 360deg

′

Prime: arcminute, 1° = 60′

″

Double prime: arcsecond, 1′ = 60″

AB

Line segment: line from point A to point B

⊥

Perpendicular: perpendicular lines (90° angle)

∥

Parallel: parallel lines

≅

Congruent to: equivalence of geometric shapes and size

~

Similarity: same shapes, not same size

Δ

Triangle: triangle shape

|x-y|

Distance: distance between points x and y

π

Pi constant: π = 3.141592654... is the ratio between the circumference and diameter of a circle

rad

Radians: radians angle unit

c

Radians: radians angle unit

grad

Gradians / gons: grads angle unit

g

Gradians / gons: grads angle unit

ALGEBRA SYMBOLS

x

Variable x: unknown value to find

≡

Equivalence: identical to

≜

Equal by definition

:=

Equal by definition

~

Approximately equal: weak approximation

∝

Proportional to

∞

Lemniscate: infinity symbol

≪

Much less than

≫

Much greater than

{ }

Braces: set

⌊x⌋

Floor brackets: rounds number to lower integer

⌈x⌉

Ceiling brackets: rounds number to upper integer

x!

Exclamation mark: factorial

|x|

Vertical bars: absolute value or modulus

f(x)

Function of x: maps values of x to f(x)

(f∘g)

Function composition: (f∘g) (x) = f(g(x))

(a,b)

Open interval: (a,b) = {x | a < x < b}

[a,b]

Closed interval: [a,b] = {x | a ≤ x ≤ b}

∆

Delta: change / difference

∆

Discriminant: Δ = b²-4ac

Σ

Sigma: summation / sum of all values in range of series

ΣΣ

Double sigma: double summation

∏

Capital pi: product / product of all values in range of series

e

Constant e / Euler's number: e = 2.718281828...

γ

Euler-Mascheroni constant: γ = 0.5772156649...

φ

Golden ratio constant

LINEAR ALGEBRA SYMBOLS

·

Dot: scalar product

×

Cross: vector product

A⊗B

Tensor product: tensor product of A and B

[ ]

Brackets: matrix of numbers

( )

Parentheses: matrix of numbers

| A |

Determinant: determinant of matrix A

det(A)

Determinant: determinant of matrix A

|| x ||

Double vertical bars: norm

A^T

Transpose: matrix transpose

A†

Hermitian matrix: matrix conjugate transpose

A*

Hermitian matrix: matrix conjugate transpose

A^-1

Inverse matrix: A A^-1 = I

rank(A)

Matrix rank: rank of matrix A

dim(U)

Dimension: dimension of matrix A

STATISTICS SYMBOLS

P(A)

Probability function: probability of event A

P(A ⋂ B)

Probability of events intersection: probability of events A and B

P(A ⋃ B)

Probability of events union: probability of events A or B

P(A | B)

Conditional probability function: probability of event A given event B occured

f(x)

Probability density function (pdf): P(a ≤ x ≤ b) = ∫ f(x) dx

F(x)

Cumulative distribution function (cdf): F(x) = P(X ≤ x)

μ

Population mean: mean of population values

E(X)

Expectation value: expected value of random variable X

E(X | Y)

Conditional expectation: expected value of random variable X given Y

var(X)

Variance: variance of random variable X

σ²

Variance: variance of population values

std(X)

Standard deviation: standard deviation of random variable X

σ_X

Standard deviation: standard deviation of random variable X

cov(X,Y)

Covariance: covariance of random variables X and Y

corr(X,Y)

Correlation: correlation of random variables X and Y

ρ_X,Y

Correlation: correlation of random variables X and Y

∑

Summation: sum of all values in range of series

∑∑

Double summation

Mo

Mode: value that occurs most frequently in population

MR

Mid-range: MR = (x_max+x_min)/2

Md

Sample median: half the population is below this value

Q₁

Lower / first quartile: 25% of population are below this value

Q₂

Median / second quartile: 50% of population are below this value = median of samples

Q₃

Upper / third quartile: 75% of population are below this value

x

Sample mean: average / arithmetic mean

s²

Sample variance: population samples variance estimator

s

Sample standard deviation: population samples standard deviation estimator

z_x

Standard score: z_x = (x-x) / s_x

X~

Distribution of X: distribution of random variable X

N(μ,σ²)

Normal distribution: gaussian distribution

U(a,b)

Uniform distribution: equal probability in range a,b

exp(λ)

Exponential distribution: f(x) = λe^-λx , x ≥ 0

gamma(c,λ)

Gamma distribution: f(x) = λcx^c-1e^-λx / Γ(c), x ≥ 0

χ²(k)

Chi-square distribution: f(x) = x^k/2-1e^-x/2 / (2^k/2 Γ(k/2))

F(k₁, k₂)

F distribution

Bin(n,p)

Binomial distribution: f(k) = _nC_k p^k(1-p)^n-k

Poisson(λ)

Poisson distribution: f(k) = λ^ke^-λ / k!

Geom(p)

Geometric distribution: f(k) = p(1-p)^k

HG(N,K,n)

Hyper-geometric distribution

Bern(p)

Bernoulli distribution

COMBINATORICS SYMBOLS

n!

Factorial - n! = 1⋅2⋅3⋅...⋅n

_nP_k

Permutation: _nP_k = n! / (n-k)!

_nC_k

Combination: _nC_k = n! / k!(n-k)!

SET THEORY SYMBOLS

{ }

Set: a collection of elements

A ∩ B

Intersection: objects that belong to set A and set B

A ∪ B

Union: objects that belong to set A or set B

A ⊆ B

Subset: A is a subset of B; set A is included in set B

A ⊂ B

Proper subset / strict subset: set A is a subset of set B, and set A is not equal to set B

A ⊄ B

Not subset: set A is not a subset of set B

A ⊇ B

Superset: A is a superset of B; set A includes set B

A ⊃ B

Proper superset / strict superset: A is a superset of B, and A is not equal to B

A ⊅ B

Not superset: set A is not a superset of set B

A ⊅ B

Not superset: set A is not a superset of set B

2^A

Power set: all subsets of A

P(A)

Power set: all subsets of A

A = B

Equality: both sets have the same members

A^c

Complement: all the objects that do not belong to set A

A \ B

Relative complement: objects that belong to A and not to B

A - B

Relative complement: objects that belong to A and not to B

A ∆ B

Symmetric difference: objects that belong to A or B but not to their intersection

A ⊖ B

Symmetric difference: objects that belong to A or B but not to their intersection

a ∈ A

Element of, belongs to: set membership

x ∉ A

Not element of: no set membership

(a,b)

Ordered pair: collection of 2 elements

A×B

Cartesian product: set of all ordered pairs from A and B

|A|

Cardinality: the number of elements of set A

#A

Cardinality: the number of elements of set A

|

Vertical bar: such that

N₀

Aleph-null: infinite cardinality of natural numbers set

N₁

Aleph-one: cardinality of countable ordinal numbers set

Ø

Empty set: Ø = { }

U

Universal set: set of all possible values

ℕ₀

Natural numbers with zero: ℕ₀ = {0, 1, 2, 3, 4,...}

ℕ₁

Natural numbers without zero: ℕ₁ = {1, 2, 3, 4, 5,...}

ℕ^*

Natural numbers without zero: ℕ^* = {1, 2, 3, 4, 5,...}

ℤ

Integer numbers: ℤ = {...-3, -2, -1, 0, 1, 2, 3,...}

ℚ

Rational numbers: ℚ = {x | x = a/b, a,b ∈ ℤ}

ℝ

Real numbers: ℝ = {x | -∞ < x < +∞}

ℝ

Extended real numbers: ℝ = {x | -∞ ≤ x ≤ +∞}, ℝ = ℝ ∪ {-∞,+∞}

ℂ

Complex numbers: ℂ = {z | z = a+bi, -∞ < a < +∞, -∞ < b < +∞}

LOGIC SYMBOLS

⋅

And

^

Caret / circumflex: and

∧

Caret: and

&

Ampersand: and

+

Plus: or

∨

Reversed caret: or

|

Vertical line: or

x'

Single quote: not / negation

x

Bar: not / negation

¬

Not / negation

!

Exclamation mark: not / negation

⊕

Circled plus / oplus: exclusive or / xor

~

Tilde: negation

⇒

Implies

⇔

Equivalent: if and only if (iff)

↔

Equivalent: if and only if (iff)

∀

For all

∃

There exists

∃!

There exists one and only one

∄

There does not exist

∴

Therefore

∵

Because / since

∵

Because / since

CALCULUS & ANALYSIS SYMBOLS

lim_x→af(x)

Limit value of a function

ε

Epsilon: represents a very small number, near zero

y'

Derivative: Lagrange's notation

y''

Second derivative: derivative of derivative

y''

Second derivative: derivative of derivative

y⁽ⁿ⁾

Nth derivative: n times derivation

dy

——

dx

Derivative: Leibniz's notation

d²y

——

dx²

Derivative: Leibniz's notation

dⁿy

——

dxⁿ

Nth derivative: n times derivation

D_xy

Derivative: Euler's notation

D_x²y

Second derivative: derivative of derivative

∂f(x,y)

———

∂x

Partial derivative

∫

Integral: opposite to derivation

∫∫

Double integral: integration of function of 2 variables

∫∫∫

Triple integral: integration of function of 3 variables

∮

Closed contour / line integral

∯

Closed surface integral

∰

Closed volume integral

i

Imaginary unit: i ≡ √-1

z*

Complex conjugate: z = a+bi → z* = a-bi

z

Complex conjugate: z = a+bi → z = a-bi

Re(z)

Real part of a complex number: z = a+bi → Re(z) = a

Im(z)

Imaginary part of a complex number: z = a+bi → Im(z) = b

| z |

Absolute value / modulus / magnitude of a complex number: |z| = |a+bi| = √(a²+b²)

arg(z)

Argument of a complex number: the angle of the radius in the complex plane

∇

Nabla / del: gradient / divergence operator

x * y

Convolution: y(t) = x(t) * h(t)

δ

Delta function

GREEK ALPHABET LETTERS

Α - α - Alpha

Β - β - Beta

Γ - γ - Gamma

Δ - δ - Delta

Ε - ε - Epsilon

Ζ - ζ - Zeta

Η - η - Eta

Θ - θ - Theta

Ι - ι - Iota

Κ - κ - Kappa

Λ - λ - Lambda

Μ - μ - Mu

Ν - ν - Nu

Ξ - ξ - Xi

Ο - ο - Omicron

Π - π - Pi

Ρ - ρ - Rho

Σ - σ - Sigma

Τ - τ - Tau

Υ - υ - Upsilon

Φ - φ - Phi

Χ - χ - Chi

Ψ - ψ - Psi

Ω - ω - Omega

OTHER SYMBOLS

→

Rightwards arrow: lim_x→a

←

Leftwards arrow

↔

Left-rigt arrow: exchange of elements

↦

Rightwards arrow from bar: associate

|

Vertical bar: such that

:

Colon: such that

~

Tilde: asymptotically equal

≡

Triple bar: coincident

≈

Squiggly equal: approximately equal

∨ = | = OR = ∪ = union

∧ = & = AND = ∩ = intersection